Strictly increasing bounded function of class $C^1$

Question

Let $f:\mathbb{R} \to \mathbb{R}$ be a strictly increasing bounded function of class $C^1$. Prove that there exists a sequence $\{x_n\}_n$ of real numbers such that $x_n\to\infty$ and $\lim_{n \to \infty} f'(x_n)=0$.

Then construct a strictly increasing bounded function $f:\mathbb{R} \to \mathbb{R}$ of class $C^1$ such that the $\lim_{x \to \infty} f' (x)$ does not exist.

I know that if we assume $a_n$ increases ($a_{n+1}\ge a_n$), then either it is bounded, or not. If yes, then it converges to the $\sup a_n$, else it goes to $+\infty$. Since it is bounded here it goes to a limit and so eventually it approaches the limit and derivative approaches 0. But I dont really know how to prove it rigorously and construct such a function such that limit DNE.

Answer 1

By the Mean Value Theorem, for any $n\in\mathbb{N}$ there is $x_n\in (n,n+1)$ such that $$f'(x_n)=f(n+1)-f(n).$$ The sequence $(x_n)_n$ is strictly increasing and goes to $+\infty$. Then $$\lim_{n\to \infty }f'(x_n)=\lim_{n\to \infty }(f(n+1)-f(n))=\lim_{n\to \infty}f(n+1)-\lim_{n\to \infty }f(n)=M-M=0$$ where $M=\sup\{f(x):x \in \Bbb{R}\}$.

Hint for the second part. Consider the continuous piecewise function $f$ defined in $\mathbb{R}$ which is $e^{x}$ for $x\in (-\infty,0]$, it is linear with slope $1$ in each interval $[n,n+\frac{1}{2^n}]$ and it is linear with slope $\frac{1}{2^n}$ in each interval $[n+\frac{1}{2^n},n+1]$ for any $n\in\mathbb{N}$. The function $f$ is strictly increasing and bounded (why?). Moreover, $f'(x)$ attains the value $1$ and values $<1/2$ infinite times as $x$ goes to infinity, and therefore $\lim_{x \to \infty} f' (x)$ does not exist. Note that $f$ is not $C^1$, but it can be made smooth by changing it suitably around the joint points.

Answer 2

An increasing and bounded from above function $f:\Bbb{R} \to \Bbb{R}$ has always a limit as $x \to +\infty$ which is $L=\sup\{f(x):x \in \Bbb{R}\}$